water flow through unsaturated porous media with hysteresis

Nonlinear Analysis: Real World Applications 12 (2011) 3125–3134

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Water flow through unsaturated porous media with hysteresisP. Kordulová ∗

Mathematical Institute, Silesian University in Opava, 746 01 Opava, Czech Republic

a r t i c l e i n f o

Article history:Received 26 June 2009Accepted 9 May 2011

Keywords:Preisach hysteresis operatorParabolic equation

a b s t r a c t

The paper is devoted to the investigation of a parabolic equationwith the Preisach operatorunder the time derivative. The model equation appears in the context of soil waterhysteresis. Under suitable assumptions an existence result is obtained by using an implicittime discretization scheme, a priori estimates and passage to the limit in the convexitydomain of the Preisach operator.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

We deal with the parabolic equation with hysteresis

∂v

∂t− u = f , v = W(u) inΩ × (0, T ), (1.1)

where Ω is an open bounded set in R2 with Lipschitzian boundary and W is a Preisach operator. This equation is coupledwith initial conditions and homogeneous Dirichlet boundary condition.

This equation can be presented as themodel of flow through unsaturated porous media. Soil–moisture hysteresis, whichis constructed by Preisach operators, is important in terrestrial hydrology, agronomy and soil physics.

Seventy-nine years ago, Haines [1] observed the presence of hysteresis in the relationship between the water contentin the soil and the water potential. For further information about the history and descriptions of soil–moisture hysteresis,see [2–6] and further references therein.

Under appropriate conditions on the initial data we investigate the existence of solution of the problem coupledwith Preisach operator under the time derivative. We show the existence result of the time discrete scheme using theBrowder–Minty fixed point theorem.We derive a priori bounds independent of the discretization parameter using a discreteversion of the second order energy inequality. This means that our solution is in the domain where all hysteresis loops areconvex, i.e., in the convexity domain. For the Preisach model this condition is satisfied only by small hysteresis loops. Thuswe can get solutions only for small initial data not to leave the Preisach convexity domain.

We apply here an idea used for proving the existence result of the model equation for magnetohydrodynamic flow [7].The paper is organized as follows. In Section 2, we briefly introduce the concept of model equation. In Section 3, we recall

somebasic definitions andproperties about play and Preisach hysteresis operators. In Section 4, themain result is postulated.The proof of the existence result is led through three steps: approximation, a priori estimates and limit procedure.

∗ Fax: +420 553 684 680.E-mail address: [email protected].

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3126 P. Kordulová / Nonlinear Analysis: Real World Applications 12 (2011) 3125–3134

2. Model equation

Flows through unsaturated porous media can be described by three conditions:

1. a water balance, that states that water mass is conserved,2. a non-linear potential flow obeying a generalized Darcy law,3. a rate independent relationship between water content and potential which closes the system of equations that express

the first two conditions.

We consider the differential equation

ρ∂Θ

∂t= −

∂h∂x, Θ(0) = Θ0, 0 ≤ Θ ≤ Θs < 1, (2.1)

where ρ is the density of water, h is the flux density;Θs is the volumetric water content at natural saturation andΘ is thevolumetric moisture content of the medium. When the soil matrix is perfectly dried then Θ = 0, when the matrix is fullysaturated with water, thenΘ = Θs < 1 and finally for in between states the matrix contains both air and water. HenceΘis bounded between 0 andΘs as indicated in Eq. (2.1).

For h we use a form of Darcy law [8] for the flow of a liquid phase through the homogeneous porous media, which wecan write as follows

h = −k∂ψ

∂x, (2.2)

where ψ is the matric potential and k is the hydraulic conductivity of the medium.Combining Eqs. (2.1)–(2.2) and eliminating the flux, we get the equation

ρ∂Θ

∂t=∂

∂x

k∂ψ

∂x

. (2.3)

The Preisach operator models the relationship between matric potential and volumetric moisture content, i.e., Θ(x, t) =

W(ψ)(x, t).

3. Hysteresis operators

Hysteresis is a phenomenon in which the response of a physical system to an external influence depends not only on thepresent magnitude of the influence but also on the previous history of the system. Hysteresis operators are characterizedby two main properties — memory effect and rate independence.

The memory effect means that at any instant t the value of the output v(t) is not simply determined by the value u(t) ofthe input at the same instant t , but it depends also on the previous evolution of the input u.

On the other hand we also require that the path of the input–output couple (u(t), v(t)) is invariant with respect to anyincreasing timehomeomorphism, i.e., there is nodependence on thederivative ofu. This property is named rate independenceand it is this fact that allows us to draw the characteristic pictures of hysteresis in the (u, v)-plane, if this did not hold wecould not give a graphic representations of the hysteresis loop as the path of the couple would depend on its velocity.

Themodernmathematical concept of hysteresiswas suggested by Krasnosel’skii and his co-workers [9]. See also [10–13].

3.1. The play operator

Firstly we briefly mention definition and properties of the classical play operator. Definition is given in this way:for a given input function u ∈ C([0, T ]) and initial condition x0r ∈ [−r, r], we define the output ξ := Pr [x0r , u] ∈

C([0, T ]) ∩ BV (0, T ) of the play operator

Pr : [−r, r] × C([0, T ]) → C([0, T ]) ∩ BV (0, T )

as the solution of the variational inequality∫ T

0[u(t)− ξ(t)− y(t)] dξ(t) ≥ 0, ∀y ∈ C([0, T ]), max

0≤t≤T|y(t)| ≤ r,

|u(t)− ξ(t)| ≤ r, ∀t ∈ [0, T ],

ξ(0) = u(0)− x0r . (3.1)

P. Kordulová / Nonlinear Analysis: Real World Applications 12 (2011) 3125–3134 3127

The concept of memory in connection to hysteresis operators is related to the fact that at any instant time t , the output v(t)may depend not only on the input u(t) and the initial condition but also on the previous evolution of input value u(t).

We notice that we can associate to any r ∈ R the corresponding value x0r ; this suggests the idea of making the initialconfiguration of the play system independent of the initial conditions x0r r>0 for the output function by the introduction ofsome suitable function of r . More precisely, following [11], Section II.2, let us consider any function λ ∈ Λwhere

Λ :=

λ ∈ W 1,∞(0,∞);

dλ(r)dr

≤ 1 a.e. in [−r, r].

We also introduce subspaces ofΛ, i.e.,

ΛK := λ ∈ Λ; λ(r) = 0 for r ≥ K, Λ0 :=

K>0

ΛK . (3.2)

Λ is called configuration space and the functions λ are calledmemory configurations.If Qr : R → [−r, r] is the projection

Qr(x) := sign(x)minr, |x| = minr,max−r, x,

then we set

x0r := Qr(u(0)− λ(r)).

This implies that the initial configuration of the play system depends on λ and on u(0). So we can introduce the followingmore convenient notation

pr [λ, u] := Pr [x0r , u], (3.3)

for any λ ∈ Λ, u ∈ C([0, T ]) and r > 0.For the sake of completeness p0[λ, u] = u. Moreover, the operator pr : Λ×C([0, T ]) → C([0, T ]) is Lipschitz continuous

in the following sense (see [11], Section II.2, Lemma 2.3).

Lemma 3.1. For every u, w ∈ C([0, T ]), λ,µ ∈ Λ and r > 0 we have

|pr [λ, u] − pr [µ,w]|∞ ≤ max|λ(r)− µ(r)|, ‖u − w‖∞.

The introduction of the function λ plays an important role in the characterization of the memory of the play system, in thesense that, for any given λ, we can construct the play operator pr [λ, u] starting from λ and from a sequence of values (tj, rj)which is so called memory sequence (see [11], Section II.2 or [13], Section III.6) of any input u at a certain instant t withrespect to the initial configuration λ. These values are what one simply has to know in order to evaluate the output of theplay operator.

In [14], the play operator is defined in the space GR(0, T ) of right-continuous regulated functions. This is the space offunctions u : [0, T ] → R which admits the left limit u(t−) at each point t > 0 and the right limit u(t+) exists and coincideswith u(t) at each point t ≥ 0. The space GR(0, T ) is endowed with the norm

‖u‖[0,T ] = sup|u(τ )|; τ ∈ [0, T ] for u ∈ GR(0, T ) (3.4)

so the GR(0, T ) is a Banach space. By Theorem 2.1 and Proposition 2.4 of [14], this is Lipschitz continuous in the sense that

|pr [λ, u](t)− pr [µ,w](t)| ≤ max|λ(r)− µ(r)|, ‖u − w‖[0,T ], (3.5)

for any λ,µ ∈ Λ, u, w ∈ GR(0, T ) and t ∈ [0, T ]. For step functions u ∈ GR(0, T ) of the form

u(t) =

m−k=1

uk−1χ[tk−1,tk )(t)+ umχT (t), (3.6)

where 0 = t0 < t1 < · · · tm = T is a given division of [0, T ], we have in particular

pr [λ, u](t) =

m−k=1

ξk−1(r)χ[tk−1,tk )(t)+ ξm(r)χT (t), (3.7)

where χω(t) is the characteristic function of a set ω ⊂ [0, T ], and

ξ0(r) = P [λ, u0](r), ξk(r) = P [ξk−1, uk](r), (3.8)


with P : Λ× R → Λ defined as

P [λ,w](r) = maxw − r,minw + r, λ(r). (3.9)

3.2. The Preisach operator

Now we briefly recall definition and some basic properties of the Preisach operator [15]. More information about thePreisach operator can be found in [9–13,16,17].

Let us introduce the Preisach half-plane, defines as

R2+

:= (r, w) ∈ R2: r > 0 (3.10)

and assume that a function ψ ∈ L1loc(R2+) (the Preisach density) is given with the following property.

Assumption 3.1. There exist β1 ∈ L1loc(0,∞), such that

0 ≤ ψ(r, w) ≤ β1(r) for a.e. (r, w) ∈ R2+.

We put

b1(K) :=

∫ K

0β1(r) dr for K > 0 (3.11)

and

g(r, w) :=

∫ w

0ψ(r, z) dz for (r, w) ∈ R2

+. (3.12)

We define the Preisach operator as follows.

Definition 3.1. Letψ ∈ L1loc(R2+) be given and let g be as in (3.12). Then the Preisach operatorW : Λ0×GR(0, T ) → GR(0, T )

generated by the function g is defined by the formula

W[λ, u](t) :=

∫∞

0g(r, pr [λ, u](t)) dr =

∫∞

0

∫ pr [λ,u](t)

0ψ(r, z) dz dr (3.13)

for λ ∈ Λ0, u ∈ GR(0, T ) and t ∈ [0, T ].

Then we have the following result (see [11], Section II.3, Proposition 3.11).

Proposition 3.1. Let Assumption 3.1 be satisfied and let K > 0 be given. Then for every λ,µ ∈ ΛK and u, w ∈ GR(0, T ) suchthat ‖u‖[0,T ], ‖w‖[0,T ] ≤ K, the Preisach operator (3.13) satisfies

‖W[λ, u] − W[µ,w]‖ ≤

∫ K

0|λ(r)− µ(r)|β1(r) dr + b1(K)‖u − w‖[0,T ] ∀t ∈ [0, T ].

We introduce the Preisach potential energy U as

U[λ, u] :=

∫∞

0G(r, pr [λ, u]) dr, (3.14)

where

G(r, w) := wg(r, w)−

∫ w

0g(r, z) dz =

∫ w

0zψ(r, z) dz, (3.15)

and the dissipation operator is given by

D[λ, u] :=

∫∞

0rg(r, pr [λ, u]) dr. (3.16)

We recover the following result (see [11], Section II.4, Theorem 4.3).

Proposition 3.2. Let the Preisach operator W satisfy Assumption 3.1 and let K > 0 be given. For arbitrary λ ∈ ΛK andu ∈ W 1,1(0, T ) such that ‖u‖C([0,T ]) ≤ K, we put

v := W[λ, u] U := U[λ, u] D := D[λ, u].

Then we have


(i) U(t) ≥1

2b1(K)v2(t) ∀t ∈ [0, T ],

(ii) v(t)u(t)− U(t) = |D(t)| a.e.

The following result can be found in [11], Section II.4, Proposition 4.8.

Proposition 3.3. Let Assumption 3.1 be satisfied and let K > 0 be given. Suppose moreover b ≥ 0, λ ∈ ΛK , and u ∈ W 1,1(0, T )be given such that ‖u‖C([0,T ]) ≤ K. Put v := bu + W[λ, u]. Then

bu2(t) ≤ v(t)u(t) ≤ (b + b1(K))u2(t). (3.17)

In Eq. (1.1), both the input function and the initial memory configuration depend on the space variable x. If λ(x, ·) belongstoΛ0 and u(x, ·) belongs to C([0, T ]) for (almost) every x, then we can define

W[λ, u](x, t) :=

∫∞

0g(r, pr [λ(x, ·), u(x, ·)](t)) dr. (3.18)

In the following we will often write W(u) instead of W[λ, u] for brevity or when λ is clear from the context.We conclude this subsection with the convexification of the Preisach operator, i.e., that in a certain region, the convexity

of the loops is satisfied (see [11], Section II.4, Proposition 4.22).Let R > 0 be fixed, set

DR := (r, w) ∈ R2+

: |w| + r ≤ R.

In addition to Assumption 3.1 we prescribe the following conditions.

Assumption 3.2. 1. ∂ψ

∂w∈ L∞

loc(R2+);

2. AR := infψ(r, w); (r, w) ∈ DR > 0.

Furthermore, denote

CR := sup ∂∂wψ(r, w)

; (r, w) ∈ DR

.

Taking possibly a smaller R > 0, if necessary, we may assume that

KR :=12AR − RCR > 0. (3.19)

We modify the density ψ outside DR and set

ψR(r, w) =

ψ(r, w) if (r, w) ∈ DR,ψ(r,−R + r) ifw < −R + r, r ≤ R,ψ(r, R − r) ifw > R − r, r ≤ R,ψ(R, 0) if r > R.

(3.20)

We define the convexified Preisach operator WR by the formula

WR[λ, u](t) =

∫∞

0

∫ pr [λ,u](t)

0ψR(r, w) dw dr (3.21)

for λ ∈ Λ0 and u ∈ W 1,1(0, T ). The convex character of WR will be exploited in Section 4.

4. Main result

Consider an open bounded set of Lipschitz classΩ ⊂ R2 and set Q := Ω × (0, T ). We set V := H10 (Ω) and H := L2(Ω).

We identify the space H to its dual H ′. So we get the Hilbert triplet V ⊂ H ≡ H ′⊂ V ′, with continuous, dense and compact

injections. We introduce the linear and continuous operator A : V → V ′ as follows:

V ′⟨Au, w⟩V :=

∫Ω

∇u∇w dx, ∀u, w ∈ V ;

hence Au = −u(:= −∑N

i=1∂2u∂x2i) in D ′(Ω) (in the sense of distributions). We assume that u0(x), v0(x) ∈ L2(Ω) are given

initial conditions and f ∈ L2(0, T ; V ′) is a given function.


We want to solve the following problem.

Problem 4.1. Let us consider a Preisach operator W and let u0 ∈ L2(Ω), λ : Ω → Λ be given initial data. We search for afunction u such that u(x, 0) = u0(x) a.e. inΩ and for any ψ ∈ L2(0, T ; V ) ∩ H1(0, T ; L2(Ω))with ψ(·, T ) = 0 a.e. inΩ , wehave ∫ T

0

∫Ω

∂v

∂tψ dx dt +

∫ T

0

∫Ω

∇u∇ψ dx dt =

∫ T

0V ′⟨f , ψ⟩V dt. (4.1)

Interpretation. The variational equation (4.1) yields

∂v

∂t+ Au = f in D ′(0, T ; V ′) (4.2)

and (4.2) holds in V ′ a.e. in (0, T ). Hence, integrating by parts in time in (4.1), we get

W(u)|t=0 = v0 in V ′ (in the sense of traces).

Now we are ready to state and prove the following existence result.

Theorem 4.1 (Existence). Let us assume operator W be the Preisach operator introduced in (3.18) and satisfying Assumptions 3.1and 3.2. And let R > 0 be fixed as in Section 3.2. Let K ∈ [0, R] and λ : Ω → ΛK be given. Moreover f ∈ W 1,2(0, T ; V ′), u0 ∈ V ,u0 ∈ L2(Ω), v0 ∈ L2(Ω) and set α := max‖u0‖V , ‖u0‖L2(Ω), ‖f ‖W1,2(0,T ;V ′). Then there exists a constant β > 0 such thatif α ≤ β , then Problem 4.1 has at least one solution such that

u ∈ L∞(0, T ;W 2,2(Ω)),

ut ∈ L2(0, T ; V ),vt ∈ L∞(0, T ;W 1,2(Ω)).

Proof. (i) Approximation. Let us fix a time step τ :=Tm , for some m ∈ N . Let us consider f0(x) = f (x, 0) and fk(x) =

1τ

kτ(k−1)τ f (x, t) dt for any k = 1, . . . ,m, a.e. inΩ. Furthermore we consider for k = 1, . . . ,m a recurrent system with the

unknown uk,

1τ

∫Ω

(vk − vk−1)ψ dx +

∫Ω

∇uk∇ψ dx =

∫Ω

fkψ dx. (4.3)

Here we set

vk(x) =

∫∞

0gR(r, ξk(x, r)) dr (4.4)

with

gR(r, w) =

∫ w

0ψR(r, w′) dw′, (4.5)

where ψR is function introduced in (3.20), i.e., we are working with the convexified Preisach operator WR(u). The sequenceξk is defined recursively by

ξ0(x, r) := P[λ(x, ·), u0(x)](r), ξk(x, r) := P[ξk−1(x, ·), uk(x)](r). (4.6)

Setting

u(τ )(x, t) =

m−k=1

uk−1(x)χ[(k−1)τ ,kτ)(t)+ um(x)χT (t), (4.7)

and

ξ (τ )r (x, t) =

m−k=1

ξk−1(x, r)χ[(k−1)τ ,kτ)(t)+ ξm(x, r)χT (t), (4.8)

we thus have, in the view of (3.6)–(3.9)

ξ (τ )r (x, t) = pr [λ, u(τ )](x, t). (4.9)


Now, we can rewrite (4.3) in the following way

vk − vk−1

τ+ Auk = fk in V ′. (4.10)

We construct the solution of (4.1) by induction over k. Assuming that uk−1 ∈ V is known, we define the mappingZ : V → V ′ as

V ′⟨Z(u), ψ⟩V =1τ

∫Ω

(v − vk−1)ψ dx +

∫Ω

∇u∇ψ dx,

where v(x) =

∞

0 gR(r, P[ξk−1(x, ·), u(x)](r)) dr.We can use a standard procedure to conclude that this equation has at least one solution. This can be done by using

a Browder–Minty’s theorem (see [18], Theorem 9.45). In particular it is necessary to show that Z is bounded, continuous,monotone and coercive.

(ii) A discrete first order energy inequality. We established here a discrete counterpart of Proposition 3.2. We set ξ rk (x) :=

ξk(x, r), where ξk(x, r)was introduced in (4.6). Letψ be an arbitrary function satisfying Assumption 3.1.We define a discreteversions of the Preisach potential energy U and dissipation operator D , introduced in (3.14) and (3.16) as

Uk(x) =

∫∞

0G(r, ξ rk (x))dr,

Dk(x) =

∫∞

0rg(r, ξ rk (x))dr,

with G given by (3.15). The discrete version of the first order energy inequality can be stated as follows:

(vk − vk−1)uk − (Uk − Uk−1) ≥

∫∞

0

∫ ξ rk

ξ rk−1

rψ(r, w)dwdr = |Dk − Dk−1|. (4.11)

(iii) A discrete second order energy inequality. The discrete version of the second order energy inequality can be stated asfollows: For every k = 2, . . . , n, n ∈ 1, . . . ,m and a.e. x ∈ Ω

(zk − zk−1)qk ≥12(zkqk − zk−1qk−1), (4.12)

where we set

qk :=uk − uk−1

τ, zk :=

vk − vk−1

τfor k = 1, . . . ,m;

we notice that, due to assumption, z0 = f0 + u0 = 0.The time continuous case is treated in detail in ([11], Sections II.3 and II.4).

(iv) A Priori Estimates. In the estimates below, let C denote a constant independent of α and τ . Indeed, the value of C mayvary from one formula to another.

We choose ψ = τuk in (4.3), we obtain∫Ω

(vk − vk−1)uk dx + τ

∫Ω

|∇uk|2 dx = τ

∫Ω

fkuk dx.

Hence, using (4.11), we have∫Ω

(Uk − Uk−1) dx + τ

∫Ω

|∇uk|2 dx ≤ τ

∫Ω

fkuk dx.

After summing up for k = 1, . . . , n, for every n ∈ 1, . . . ,m, using the regularity of the initial data, we obtainn−

k=1

∫Ω

(Uk − Uk−1) dx + τ

n−k=1

∫Ω

|∇uk|2 dx ≤ τ

n−k=1

∫Ω

fkuk dx,

τ

n−k=1

‖∇uk‖2L2(Ω) ≤

τ

2

n−k=1

‖fk‖2L2(Ω) +

τ

2

n−k=1

‖uk‖2V +

∫Ω

U0 dx

τ

2

n−k=1

‖uk‖2V ≤ Cα2. (4.13)


Due to the monotonicity and local Lipschitz continuity of the functions g(r, ·) and P [λ, ·](r), we have the pointwiseinequality

qk(x)zk(x) ≥ 0 a.e. inΩ, k = 1, . . . ,m. (4.14)

By taking the incremental ratio in time in (4.3), we have∫Ω

zk − zk−1

τψ dx +

∫Ω

∇qk∇ψ dx =

∫Ω

fk − fk−1

τψ dx. (4.15)

We choose ψ = τqk in (4.15). We get∫Ω

(zk − zk−1)qk dx + τ

∫Ω

|∇qk|2 dx = τ

∫Ω

fk − fk−1

τqk dx. (4.16)

Then let us sum for k = 1, . . . , n, for every n ∈ 1, . . . ,m, use (4.12) for k ≥ 2 and (4.14), we obtain

n−k=1

∫Ω

(zk − zk−1)qk dx + τ

n−k=1

∫Ω

|∇qk|2 dx

≥

∫Ω

(q1z1 − q1z0) dx +12

n−k=2

∫Ω

(qkzk − qk−1zk−1) dx + τ

n−k=1

∫Ω

|∇qk|2 dx

=

∫Ω

(q1z1 − q1z0) dx +12

∫Ω

(qnzn − q1z1) dx + τ

n−k=1

∫Ω

|∇qk|2 dx

≥ τ

n−k=1

∫Ω

|∇qk|2 dx.

On the other hand

τ

n−k=1

∫Ω

fk − fk−1

τqk dx ≤

τ

2

n−k=1

fk − fk−1

τ

2

V ′

+τ

2

n−k=1

‖qk‖2V .

From the previous two chains of inequalities, we deduce

τ

n−k=1

‖∇qk‖2L2(Ω) ≤

τ

2

n−k=1

fk − fk−1

τ

2

V ′

+τ

2

n−k=1

‖qk‖2V ,

which yields the following a priori estimate

τ

n−k=1

‖qk‖2V ≤ Cα2. (4.17)

Finally we need L∞ bound for our solution. We prove by induction over k = 1, . . . ,m that there exists B > 0 independentof k and m such that

‖uk‖L∞(Ω) ≤ Bα for all k = 0, . . . ,m. (4.18)

For k = 0, 1, . . . ,m we set

B(m)k =1α

max‖uj‖L∞(Ω); j = 0, 1, . . . , k.

We have B(m)0 ≤ C independently of m by hypothesis on initial data. Let now 1 ≤ k0 ≤ m be fixed and assume thatB(m)k0−1 < ∞. By direct comparison in Eq. (4.3), we derive for k = 1, . . . , k0 the estimate

‖uk‖L2(Ω) ≤ ‖zk‖L2(Ω) + ‖fk‖L2(Ω). (4.19)

This yields that B(m)k0< ∞. According to (3.20) and (4.4)–(4.6), we get for every k and a.e. x ∈ Ω the pointwise estimate

‖zk(x)‖ ≤ C1 + max

j=0,...,k‖uk(x)‖

‖qk(x)‖.


Hence, by definition of α and by (4.17), we obtain from (4.19) that

‖uk0‖L2(Ω) ≤ C(1 + maxB(m)k0−1α, ‖uk0‖L∞(Ω))‖qk0‖L2(Ω) + ‖fk‖L2(Ω)

≤ Cα(1 + maxB(m)k0−1α, ‖uk0‖L∞(Ω)). (4.20)

Therefore

‖uk0‖W2,2(Ω) ≤ C‖uk0‖L2(Ω) ≤ Cα(1 + maxB(m)k0−1α, ‖uk0‖L∞(Ω)). (4.21)

Hence, the embedding ofW 2,2(Ω) into W 1,4(Ω) yields,

‖∇uk0‖L4(Ω) ≤ Cα(1 + maxB(m)k0−1α, ‖uk0‖L∞(Ω)).

Using the Gagliardo–Nirenberg inequality (see e.g. [19]) in the form

‖uk0‖L∞(Ω) ≤ C‖uk0‖13L2(Ω)‖∇uk0‖

23L4(Ω)

and (4.13), we obtain that ‖uk0‖L∞(Ω) ≤ Cα(1 + maxB(m)k0−1α, ‖uk0‖L∞(Ω))23 . Assume that B(m)k0

> B(m)k0−1. Then

B(m)k0=

1α

max‖uk0‖L∞(Ω) ≤ C(1 + α0B(m)k0)23 ,

hence B(m)k0≤ maxC, B(m)k0−1 with a constant C independent of k and m, and the desired estimate (4.18) follows. Inequality

(4.20) implies in particular that

‖uk‖L2(Ω) ≤ Cα for all k = 1, . . . ,m. (4.22)

(v) Limit procedure. With the sequence uk constructed above (see (4.7)–(4.9)) we define for each fixed time step τ thefunctions

u(τ )+ (x, t) = uk(x), u(τ )− (x, t) = uk−1(x),

v(τ )+ (x, t) = vk(x), f (τ )+ (x, t) = fk(x),

(4.23)

and u(τ )(x, t) = uk−1(x)+

t − (k − 1)ττ

(uk(x)− uk−1(x))

v(τ )(x, t) = vk−1(x)+t − (k − 1)τ

τ(vk(x)− vk−1(x))

(4.24)

for every x ∈ Ω and t ∈ [(k − 1)τ , kτ), k = 1, 2, . . . ,m, continuously extended to t = T . We have

v(τ )+ = WR[λ, u

(τ )+ ]. (4.25)

As a consequence of the estimates (4.17), (4.22) and by comparison of the terms of the Eq. (4.3) we get that u ∈

L∞(0, T ;W 2,2(Ω)∩V ), ut ∈ L2(0, T ; V ) and vt ∈ L∞(0, T ;W 1,2(Ω)). The a priori estimates we found allow us to concludethat there exists u, ut and vt such that, along a subsequence as τ → 0, we have

u(τ ) → u weakly star in L∞(0, T ;W 2,2(Ω)),

ut(τ )

→ ut weakly star in L2(0, T ; V ),vt(τ )

→ vt weakly star in L∞(0, T ;W 1,2(Ω)).

(4.26)

By compact embedding, we have, passing again to a subsequence, if necessary

∇u(τ ) → ∇u strongly in L2(Q ; R2),

u(τ ) → u uniformly in C0(Q ).(4.27)

We further have for every τ and every (x, t) ∈ Q that

|u(τ )(x, t)− u(τ )± (x, t)|2 ≤ maxk

|uk(x)− uk−1(x)|2 ≤

m−k=1

|uk(x)− uk−1(x)|2,

|v(τ )(x, t)− v(τ )+ (x, t)|2 ≤ max

k|vk(x)− vk−1(x)|2 ≤ C

m−k=1

|uk(x)− uk−1(x)|2.


From (4.17) it follows that

‖u(τ ) − u(τ )± ‖L2(Ω;GR(0,T )) + ‖v(τ ) − v(τ )+ ‖L2(Ω;GR(0,T )) ≤ C

√τ , (4.28)

‖∇u(τ ) − ∇u(τ )± ‖L2(Q ;R2) ≤ C√τ . (4.29)

Hence u(τ )± converge strongly to u in L2(Ω;GR(0, T )) as τ → 0. By Proposition 3.1 we may pass to the limit in (4.25) andobtain

v(τ )+ → v = WR[λ, u] strongly in L2(Ω;GR(0, T )). (4.30)

This and (4.28) yield

v(τ ) → v strongly in L2(Ω;GR(0, T )). (4.31)

Eq. (4.3) is in the form∫Ω

v(τ )t ψ dx +

∫Ω

∇u(τ )+ ∇ψ dx =

∫Ω

f (τ )+ ψ dx. (4.32)

The convergences (4.26)–(4.27), (4.30)–(4.31) and inequality (4.29) enable us to pass to the limit as τ → 0 and obtain∫Ω

(vtψ + ∇u∇ψ) dx =

∫Ω

fψ dx. (4.33)

At this point we can deduce that there exists a constant β depending on R such that if α ≤ β then |u(x, t)| ≤ R a.e. inQ . This implies that W(u) = WR(u) (see Lemma II.2.4 in [11] and the definition of ψR (3.20)). This completes the proof oftheorem.

Acknowledgments

The research was supported, in part, by the Grant Agency of the Czech Republic, grant No. 201/09/P163 and by projectMSM4781305904 from the Czech Ministry of Education.

References

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water flow through unsaturated porous media with hysteresis

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